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Suppose you throw a six-sided die five times. Find the probability that the sum of the outcomes of the throws is 15 using generating functions. Here's my attempt:
Okay, so I know so far that the generating function of a single throw is
G(x) = s/6 + (s^2)/6 + (s^3)/6 + (s^4)/6 + (s^5)/6) + (s^6)/6.
And that G(x) raised to the 5th power is the generating function for five throws.
Also, taking into account the independence of the x's:
G_x(t) = ((s+...+s^6)/6)^5 = s^5 * (1-s^6)^5 divided by 6^5 * (1-s)^5.
I get that:
1/(1-s)^5 = 1 + (5 choose 1) s + (6 choose 2) s^2 + (7 choose 3) s^3 + (8 choose 3) s^4 + (9 choose 5) s^5
and that
(1-s^6)^5 = 1 - (5 choose 1) s^6 + (5 choose 2) s^12 - (5 choose 3) s^18 + (5 choose 4) s^24 - (5 choose 5) s^30
However, I'm confused about how I use these to figure out the coefficient of s^15, which is the probability I'm looking for.
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